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Tetrisation of Triangular Meshes and Its Application in Shape Blending

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Book cover Mathematical Progress in Expressive Image Synthesis III

Part of the book series: Mathematics for Industry ((MFI,volume 24))

Abstract

The As-Rigid-As-Possible (ARAP) shape deformation framework is a versatile technique for morphing, surface modelling, and mesh editing. We discuss an improvement of the ARAP framework in a few aspects: 1. Given a triangular mesh in 3D space, we introduce a method to associate a tetrahedral structure, which encodes the geometry of the original mesh. 2. We use a Lie algebra based method to interpolate local transformation, which provides better handling of rotation with large angle. 3. We propose a new error function to compile local transformations into a global piecewise linear map, which is rotation invariant and easy to minimise. We implemented a shape blender based on our algorithm and its MIT licensed source code is available online.

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Notes

  1. 1.

    The term involving I is for normalisation and it enforces \(\mathrm {Blend}_P( 0,\ldots ,0, \hat{A}_{1i}, \hat{A}_{2i}, \ldots , \hat{A}_{mi} )=I\).

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Acknowledgments

This work was partially supported by the Core Research for Evolutional Science and Technology (CREST) Program titled “Mathematics for Computer Graphics” of the Japan Science and Technology Agency (JST), by KAKENHI Grant-in-Aid for Young Scientists (B) 26800043, and by JSPS Postdoctoral Fellowships for Research Abroad.

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Authors and Affiliations

  1. Yamaguchi University/JST CREST, 1677-1, Yoshida, Yamaguchi, 753-8512, Japan

    Shizuo Kaji

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  1. Shizuo Kaji
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Correspondence to Shizuo Kaji .

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Editors and Affiliations

  1. Grad. School of Info. Sci. and Tech, Hokkaido University, Sapporo, Hokkaido, Japan

    Yoshinori Dobashi

  2. Kyushu University, Fukuoka, Japan

    Hiroyuki Ochiai

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© 2016 Springer Science+Business Media Singapore

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Kaji, S. (2016). Tetrisation of Triangular Meshes and Its Application in Shape Blending. In: Dobashi, Y., Ochiai, H. (eds) Mathematical Progress in Expressive Image Synthesis III. Mathematics for Industry, vol 24. Springer, Singapore. https://doi.org/10.1007/978-981-10-1076-7_2

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  • DOI: https://doi.org/10.1007/978-981-10-1076-7_2

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-1075-0

  • Online ISBN: 978-981-10-1076-7

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